Antimatter is made up of antiparticles. Whereas ordinary matter is made from atoms, which are made from nucleons, made from quarks, antimatter is made from antiatoms, made from antinucleons, made from antiquarks. Whenever an antiparticle meets a particle, the two annihilate into a burst of radiation; for this reason antiparticles almost never exist for long enough to form an antiatom. However, they still play an important role in the world of particle interactions.
The existence of antiparticles was predicted in 1927 by the British physicist Paul Dirac. He was attempting to combine quantum mechanics, the theory which explains the behaviour of particles on a very small scale, with Albert Einstein's theory of special relativity, which explains the behaviour of particles moving at very high speeds.
Quantum mechanics predicts that a particle can only exist in certain energy states with certain discrete amounts of energy. Relativity predicts that the energy of a particle at high speed is given by its kinetic energy, plus its rest mass energy: E=mc2. On calculating the total energy of this particle, it turns out that the energy is given by the square root of an expression involving its mass and momentum.
The square root was a problem, as every square root has both a positive and negative solution1. This suggested that a particle could have negative energy. As no physicist had ever heard of negative energy, most would simply sweep the negative energy result under the carpet. However, when combined with quantum mechanics, it suggested the existence of negative energy states.
Dirac proposed that all the negative energy states were occupied by negative energy particles, and when there is sufficient energy around (as radiation), a negative energy particle can be excited into a positive energy state. This creates a positive energy particle and an empty negative energy state. This empty state or hole can move through space just like a particle, but it is really the absence of a negative energy particle, so it has positive energy (and hence mass), and the opposite electrical charge - it is an antiparticle. If the corresponding particle falls down into the hole it will annihilate with the antiparticle and release the energy stored in their masses.
Discovery of Antiparticles
Dirac completed his calculations for electrons, and predicted the existence of an antiparticle with the mass of the electron, and the opposite charge. In 1931 this particle - an antielectron or positron - was discovered in cosmic rays. Positrons are produced fairly frequently when cosmic rays collide with the Earth's atmosphere, and when some radioactive atoms decay. However, producing heavier antiparticles - antiprotons and antineutrons - is more difficult, requiring high energy particle accelerators. The antiproton was discovered in 1955.
Ordinary atoms are made up of nucleons (protons and neutrons), and electrons. By combining the equivalent antiparticles, you could make an antiatom. However, this is an extremely difficult process; after producing antiprotons in high energy processes, you must then slow them down until they are cool enough to stick to positrons. To date, nine antiatoms of hydrogen have been produced in an experiment at CERN, the European Laboratory for Particle Physics. While nine antiatoms is not a lot, it is enough to excite physicists, and there are plans to build an antimatter factory to produce more.
A more interesting thing to do with an antiparticle is to let is it annihilate with its corresponding particle. When electrons and positrons annihilate, the energy contained in their masses is released as radiation, or photons. Other experiments at CERN accelerate electrons and positrons to very high speeds before colliding them together. At high enough energies, heavier particle-antiparticle pairs are produced.
In theory, the energy released in these annihilations could be used to fuel a spacecraft, however, to achieve this there are three main problems which would have to be overcome: how to produce a useful amount of antimatter, how to store it, and how to convert the energy from matter-antimatter annihilations into thrust to propel a starship. In practice the first problem is the most difficult; producing antimatter is an extremely inefficient process, and particle accelerators can only produce tiny quantities. Annihilating a kilo of antimatter with a kilo of matter would release the amount of energy contained in 28 million barrels of oil, but it would take a lot more than this to produce a kilo of antimatter. If direct antimatter drives are possible, we are light years away from building one.
Matter and Antimatter
From their description so far, it would appear that matter and antimatter are equal and opposite. They are perfectly symmetric, except for one small thing: the universe appears to be made almost entirely from matter. It's easy to understand why this is; any antimatter produced simply annihilates with matter. But if equal amounts of matter and antimatter were produced in the big bang (as theory predicts), then both matter and antimatter should have all annihilated long ago, and the universe should now just be filled with energy.
Therefore it seems there must be a subtle difference between matter and antimatter. The symmetry is not entirely perfect and slightly more matter than antimatter was produced at the beginning of the universe. After the universe cooled following all the annihilations, there was just enough matter left to form stars, planets, mammoths and scientists.
This small asymmetry is known to theoretical physicists as CP violation. Recent experiments have shown that it does exist, but understanding exactly what it is will require more antiparticles, and more experiments.